/ 
irrational functions. 
Remove the multiplier x n , as in Prop. V., and it becomes. 
(i 
np 
8 a Jr 
£f . X q . (a -j” @X n ) <7 . R (o 5 ( 2 >X"~~ n ') 
x 
m + 
np 
which will fall under Prop. IV. If x ? can be expressed 
by a rational function of « -f" This will happen if 
. m „ p 
q ' n * q 
know 
dx t> f ~ i 
m + ~ — — n . r, or if - + ~= ± r any integer. Hence we 
^ ^ Q 
/; 
(tfx * H- f 
f d x ^ C dx ' P 
JW~K + Ba» 'JWlfi — i 
do: 
+ j3x n V 'V** + 
Cor. 2. We can determine 
it 
OCX 
In . 
0 
, do: 
+ yx n -f (3 a/' 1 -J- ex” + /3 
which becomes by Prop. 5. 
m— — n— i 
«x -— • px 
ax + pX + 7 
dx 
X . 
n %/ n 
V /y. J? 
«/ + -f- c 
Now this falls under Prop. IV. For let 
n V ax n + (Zx— n + c = v 
CiX n 4- { 2 x— n + ry = v n + 7 — C 
;fax n ^ — ! 3 x — n ~ l ) . dx = 1 . dv 
Whence the fluxion becomes 
n 
n—z 
. dv 
, of which a particular 
it , 
•+• y • — c 
case is deduced in Legendre’s Elliptic Transcendents. 
Prop. VII. 
If we can rationalize 
dx . R) x, <p (T), (p (x), . . c 
we also can 
